Two improved extrapolated gradient algorithms for pseudo-monotone variational inequalities

1.   Introduction

Let $C$ be a nonempty closed convex subset in the real Hilbert space $H$, and let $A:H \to H$ be a mapping. The variational inequality problem is to find $x^* \in C$ that is satisfied with the following inequality:

In recent years, due to the wide application of variational inequalities in mathematics ^[1,2], physics ^[3,4,5,6], economics ^[7,8,9,10], and other fields, the theory of variational inequalities has become an important direction during the process of investigating the nonlinear analysis. Many academics have been studying it and put forward a multitude of findings ^{[11,12,13,14]}. Among the projection algorithms of variational inequalities, the Korpelevich's extragradient ^[15] algorithm is one of the most effective algorithms, that is, the following iterative formula for producing a sequence $\{x_n\}$:

in which $L$ is the Lipschitz constant of the mapping $A$ and $\lambda \in (0, \dfrac{1}{L})$. In each iteration, one needs to compute the projection onto the feasible set twice. It should be pointed out that if the structure of $C$ is complex, the projection will be difficult to calculate, which in turn impacts the algorithm's efficiency. In 2000, Tseng introduced an extragradient method with a single projection; that is, one needs to calculate only a single projection onto the feasible set during each iteration ^[16]. The specific iteration formula was given as follows:

where $L$ is the Lipschitz constant of the mapping $A$ and $\lambda \in (0, \dfrac{1}{L})$. It is worth noting that the steps of the above algorithms should all depend on the Lipschitz constant of mapping $A$. However, the Lipschitz constant is often difficult to calculate or approximate. In order to avoid estimating the Lipschitz constant, it is common to utilize the Armijo linear search to get the step size ^[17,18]. However, determining the appropriate set size can demand significant computational effort at each iteration, making the linear search with Armijo's rule rather costly. Recently, Yang and Liu proposed an adaptive external gradient algorithm ^[19], in which the step size was updated step by step by simple calculation instead of estimating the Lipschitz constant.

Based on literature ^[20,21], this paper introduces two new projection algorithms. Under the assumption that the mapping $A$ is pseudo-monotone and Lipschitz continuous, it is proved that the sequence generated by the algorithm converges strongly to the solution of the variational inequality. Compared with literature ^{[15,16,17,18,19]}, the step size of the algorithm proposed in this paper is updated step by step through simple calculation without the need to estimate the Lipschitz constant.

2.   Preliminaries

Let $H$ be a real Hilbert space, and $C \subset H$ is a nonempty closed convex subset. If $\{x_n\} \subset C$ and $\{x_n\}$ weakly converges to $x$, which is denoted as $x_n \rightharpoonup x (n \to \infty)$; $\{x_n\}$ strongly converges to $x$, which is denoted as $x_n \to x (n \to \infty)$. For all $x, y \in H$ and $\alpha \in \mathbb{R}$, we have

and

Definition 2.1. Let $H$ be a real Hilbert space, and $A:H \to H$ is a nonlinear operator, and then we call $A$ as

(1) $L-Lipschitz$ continuous $(L > 0)$ if

(2) Monotone if

(3) Pseudo-monotone if

(4) Weakly sequentially continuous if for all sequences $\{x_n\} \in H$, when $n \to \infty$, we have

Definition 2.2. ^[22] Let $H$ be a real Hilbert space, and $C \subset H$ is a nonempty closed convex subset. For all $x \in H$, there exists a unique approximation point in $C$, which is subjects to $||x-P_C(x)|| \leq ||x-y||$, $\forall y \!\in\! C$. We denote it by $P_C(x)$, and we call it a metric projection.

Lemma 2.1. ^[22] Let $H$ be a real Hilbert space, and $C \subset H$ is a nonempty closed convex subset. For all $x \in H$, $z \in C$, we have

Lemma 2.2. ^[22] Let $H$ be a real $Hilbert$ space, and $C \subset H$ is a nonempty closed convex subset, and $x \in H$, we have

(1) $||P_C(x)-P_C(y)||^2 \leq \langle P_C(x)-P_C(y), x-y \rangle, \forall y \in C$,

(2) $||P_C(x)-y||^2 \leq ||x-y||^2-||x-P_C(x)||^2, \forall y \in C$,

(3) $\langle (I-P_C)(x)-(I-P_C)(y), x-y \rangle \geq ||(I-P_C)(x)-(I-P_C)(y)||^2, \forall y \in C$.

Lemma 2.3. ^[12] For all $x \in H, \alpha \geq \beta > 0$, the following inequality holds:

(1) $||x-P_C[x-\beta A(x)]|| \leq ||x-P_C[x-\alpha A(x)]||$,

(2) $\cfrac{||x-P_C[x-\alpha A(x)]||}{\alpha} \leq \cfrac{||x-P_C[x-\beta A(x)]||}{\beta}$.

Lemma 2.4. ^[23] Let $\{x_n\} \subset (0, 1)$ be a real number sequence, $\sum\limits_{n = 1}^{\infty} \alpha_n = \infty$ and satisfy:

If $\limsup\limits_{k\to\infty}y_{n_k} \leq 0$, and for every subsequence $\{\alpha_{n_k}\}$ of $\{\alpha_n\}$, we have $\liminf\limits_{k\to\infty}(x_{{n_k}+1}-x_{n_k}) \geq 0$, then $\lim\limits_{n \to \infty}x_n = 0.$

Lemma 2.5. ^[24] Let $A:H \to H$ be $L-Lipschitz$ continuous in $H's$ bounded subset, and $M$ is a bounded subset in $H$; then $A(M)$ is bounded.

Lemma 2.6. ^[25] Let $A:C \to H$ be a pseudo-monotone continuous mapping, and $x^* \in C$, then

Lemma 2.7. ^[26] Let $\{x_{n_j}\}$ be a subsequence of the nonnegative real number sequence $\{x_n\}$, and satisfy for all $j \in \mathbb{N}$. We have $x_{n_j} < x_{{n_j}+1}$, then there exists a monotonically increasing subsequence of integers $\{m_k\}$ that subjects to $\lim\limits_{k \to \infty}m_k = \infty$. When $k \in \mathbb{N}$ is large enough, we have

In fact, $m_k$ is the largest number in the set $\{1, 2, \cdots, k\}$ and satisfies $x_n < x_{n+1}$.

Lemma 2.8. ^[27] Let $\{x_n\}$ be a sequence of nonnegative real number that satisfies

where $\{\gamma_n\}$, $\{y_n\}$ meet

then

Lemma 2.9. ^[21] Let $\{p^k\} \subset C$ be a bounded sequence. We suppose $p$ is a cluster point of $\{p^k\}$; when $\lim \limits_{k \to \infty}||p^k-p||$ exists, $\{p^k\}$ is convergent.

3.   Improved adaptive subgradient outer gradient algorithm and convergence proof

Assumpton 3.1.

(C1) Feasible set $C$ is a nonempty closed convex subset in Hilbert space $H$. Solution set $VI(C, A)$ is nonempty.

(C2) Operator $A:H \to H$ is pseudo-monotone in $C$ 's bounded set, $L-Lipschitz$ continuous, and weakly sequentially continuous.

(C3) $f:H \to H$ is a contraction mapping $(\rho \in [0, 1))$, the sequence $\{\beta_n\} \subset (0, 1]$ is satisfied with:

Next, the improved adaptive subgradient outer gradient algorithm is given.

Algorithm 3.1. Improved adaptive subgradient outer gradient algorithm.

Step 0: Given $\lambda_0 > 0$, $\mu > 2$, $\xi \in (0, 1]$. $\forall x_0, x_1 \in H$, $\{\tau_n\}$ is a positive real number sequence. $\lim\limits_{n \to \infty} \dfrac{\tau_n}{\lambda_n} = \eta$, $\eta$ is a constant, and $\eta \in (\dfrac{2}{\mu}, 1]$.

Step 1: Compute

Step 2: Compute

If $\omega_n = y_n$ or $A(y_n) = 0$, STOP. Otherwise, go to Step 3.

Step 3: Compute

Step 4: Compute

Set $n: = n+1$, and go to Step 1.

Lemma 3.1. Suppose that (C1)–(C3) hold; then $\{\lambda_n\}$ is a nondecreasing sequence generated by (3.1) and satisfies

Proof. According to (3.1), we gain that $\{\lambda_n\}$ is a nondecreasing sequence. Since $A$ is $L-Lipschitz$ continuous, we can also gain $||A(x_n)-A(y_n)|| \leq L||x_n-y_n||.$ When $\langle A(\omega_n)-A(y_n), z_n-y_n \rangle > 0$, we have

so $\{\lambda_n\}$ is nondecreasing and has an upper bound, then there exists $\lim\limits_{n \to \infty}\lambda_n = \lambda$, and $\lambda_{n+1} \leq \max\{\lambda, \dfrac{\mu L}{2}\}.$ Based on the mathematical induction, we obtain

When $\langle A(\omega_n)-A(y_n), z_n-y_n \rangle \leq 0$, we get the conclusion.

Lemma 3.2. Suppose (C1)–(C3) are set up; $\{z_n\}$ is the sequence generated by the Algorithm 3.1. Then

Proof. We know that it can be obtained that $p \in VI(C, A) \subset C \subset T_n$. According to Lemma 2.2, we have

For $y_n \in C$, we gain $\langle A(p), y_n-p \rangle \geq 0$. Because $A$ is a pseudo-monotone operator, then $\langle A(y_n), y_n-p \rangle \geq 0$. Hence

in which

Furthermore, combined with $y_n = P_C[\omega_n-\dfrac{\xi}{\tau_n}A(\omega_n)]$ and $z_n \in T_n$, we have

Substituting (3.5) into (3.4), one has

that is to say

Multiply both sides of (3.6) with $\dfrac{\tau_n}{\lambda_n}$, we have

It follows from the definition of $\{\lambda_n\}$ that

In fact, if $\langle A(\omega_n)-A(y_n), z_n-y_n \rangle \leq 0$, then the inequality (3.8) holds. Otherwise, from (3.1), it infers

which implies that (3.8) holds. In addition, combining (3.7) and (3.8), we have

Putting (3.9) into (3.3), one obtain

The proof is completed.

Lemma 3.3. Suppose (C1)–(C3) are set up; $\{\omega_n\}$ is the sequence generated by Algorithm 3.1. If there exists a subsequence $\{\omega_{n_k}\} \subset \{\omega_n\}$ weakly converging to $z \in H$ such that

then we have $z \in VI(C, A).$

Proof. According to the definition of $T_n$, we have

which equals

After expanding the right-hand side of the above equation, we can obtain

Following transposition, we have

Because $\{\omega_n\}$ weakly converges to $z$, we know $\{\omega_{n_k}\}$ is bounded. Once more, due to the $A's$ Lipschitz continuity, we have $\{A(\omega_{n_k})\}$ is bounded. As $||\omega_{n_k}-y_{n_k}|| \to 0$, $\{y_{n_k}\}$ is also bounded, and $0 < \tau_n \leq \lambda_n \leq \max\{\lambda, \dfrac{\mu L}{2}\}.$

Set $k \to \infty$ in (3.10), we gain

In addition, we have

It follows from $\lim\limits_{k \to \infty}||\omega_{n_k}-y_{n_k}|| = 0$ and the $L-$ Lipschitz continuity of $A$ in $H$, one obtain

Combining the above equation with (3.11) and (3.12), we have

Next, we prove that $z \in VI(C, A).$ Choose decreasing sequences of positive terms $\{\epsilon_k\}$ and $\liminf\limits_{k \to \infty}\epsilon_k = 0.$ For any $k$, $N_k$ denotes the smallest positive integer such that

Due to $\{\epsilon_k\}$ being decreasing, then $N_k$ is also obviously decreasing. Moreover, for any $k$, since $\{y_{N_k}\} \subset C$, we can suppose $A(y_{N_k}) \ne 0$(Otherwise, $y_{N_k}$ is the solution) and set

which infers that $\langle A(y_{N_k}), v_{N_k} \rangle = 1$. Now, it follows from (3.13) that for any $k$,

Since $A$ is pseudo-monotone, the above equation can be reduced to

which implies that

Next, we will prove that $\lim\limits_{k \to \infty}\epsilon_k v_{N_k} = 0$. Since $\omega_{n_k} \rightharpoonup z$ and $\lim\limits_{k \to \infty}||\omega_{n_k}-y_{n_k}|| = 0$, we can get $y_{N_k} \rightharpoonup z(k \to \infty)$. While $\{y_n\} \subset C$, we have $z \in C$. According to $A$, it is weakly sequence continuous in $H$, and we have $\{A(y_{N_k})\}$ weakly converges to $A(z)$. Assume that $A(z) \ne 0$(Otherwise, $z$ is the solution). Since the norm map is weakly sequential lower semicontinuous, one has

Combined with $\{y_{N_k}\} \subset \{y_{n_k}\}$ and $\epsilon_k \to 0 (k \to \infty)$, we obtain

which infers $\lim\limits_{k \to \infty}\epsilon_k v_{N_k} = 0$.

Let $k \to \infty$ in (3.13); since $A$ is continuous, and the right-hand side tends to 0, $\{\omega_{N_k}\}$ and $\{v_{N_k}\}$ are bounded, and $\lim\limits_{k \to \infty}\epsilon_k v_{N_k} = 0$. Therefore, we have

Since $\forall x \in C$, we have

From Lemma 2.6, one obtain

The proof is completed.

Theorem 3.1. Suppose that (C1)–(C3) hold, the sequence $\{\alpha_n\}$ is satisfied with:

Then the sequence $\{x_n\}$ generated by Algorithm 3.1 strongly converges to $p \in VI(C, A)$ and $p = P_{VI(C, A)} \circ f(p)$.

Proof. For the convenience of the proof, we divide it into the following four steps.

Step 1: Prove that $\{x_n\}$ is bounded.

In fact, it follows from Lemma 3.2 and the definition of $\{\tau_n\}$ that there exists $N_1 \in \mathbb{N}$ subject to

Hence

Otherwise

Since $\lim\limits_{n \to \infty}\dfrac{\alpha_n}{\beta_n}||x_n-x_{n-1}|| = 0$, there exist $N_2 \in \mathbb{N}$ and constant $M_1 > 0$ such that

Simultaneous (3.17) and (3.18), we have

Set $N = \max\{N_1, N_2\}$. Combining (3.16) and (3.19), we then gain

According to the above discussion, it can be proved that $\{x_n\}$ is bounded.

Step 2: Prove

in which $M_4 > 0$ is a constant. In fact,

and $M_2 \triangleq \sup \limits_{n \in \mathbb{N}} \{2||z_n-p|| \cdot ||f(p)-p||+||f(p)-p||^2 > 0\}$. By substituting (3.2) into (3.21), it can be obtained that

From (3.19), one has

in which $M_3 \triangleq \sup \limits_{n \in \mathbb{N}} \{2M_1||x_n-p||+\beta_n M_1^2\} > 0$. Substitute (3.23) into (3.22), we gain

which infers that

and $M_4 \triangleq M_2+M_3$.

Step 3: Prove that there exists some constant $M > 0$ such that

In fact, applying (2.1) and $\{\omega_n\}$ 's definition, we have

Combined with (2.1), (2.2), and (3.15), we obtain

Since $||\omega_n-p||^2 \leq ||x_n-p||^2+2\alpha_n||x_n-x_{n-1}|| \cdot ||\omega_n-p||$ holds, and we put (3.24) into (3.25), one has

and $M \triangleq \sup \limits_{n \in \mathbb{N}} \{||x_n-p||\} > 0.$

Step 4: Prove $\{||x_n-p||^2\}$ converges to 0.

In fact, according to Lemma 2.4, we only need to prove the sequence $\{||x_n-p||\}$ 's subsequence, which satisfies $\liminf \limits_{k \to \infty} (||x_{n_{k+1}} -p||-||x_{n_k} -p||) \geq 0$, $\{||x_{n_k} - p||\}$ meet

Thus, assume that $\{||x_{n_k} - p||\}$ is the subsequence of $\{||x_n-p||\}$ and satisfies $\liminf \limits_{k \to \infty} (||x_{n_{k+1}} -p||-||x_{n_k} -p||) \geq 0$, then

According to Step 2, we have

As the coefficients of $||\omega_{n_k}-z_{n_k}||^2, ||\omega_{n_k}-y_{n_k}||^2, ||z_{n_k}-y_{n_k}||^2$ are all positive, hence

Next, we will prove that

As $k \to \infty$, we have

and

From (3.26)–(3.28), one gets

Since $\{x_{n_k}\}$ is bounded, then there exists a subsequence $\{x_{n_{k_j}}\} \subset \{x_{n_k}\}$ weakly converging to $z \in H$, thereby

Apply (3.28), then $\omega_{n_k} \rightharpoonup z(k \to \infty)$. Combining (3.26) and Lemma 3.3, we have $z \in VI(C, A)$. Available from (3.29) and $p = P_{VI(C, A) \circ f(p)}$, we gain

Then

Hence it follows from Lemma 2.4, (3.24), and (3.31) that $\lim \limits_{n \to \infty}||x_n-p|| = 0$. The proof is completed.

Remark 3.1. The algorithm combines the subgradient outer gradient method, the inertia method, and the viscosity method. Under appropriate conditions, different parameters are introduced to improve the algorithm, and the selection of $\{\lambda_n \}$ and $\{\tau_n \}$ in the existing algorithm is improved to the reciprocal form to avoid excessive iteration and control convergence. The second class of algorithms is given next, and the convergence is proved.

4.   Golden section algorithm and convergence proof

Algorithm 4.1. Golden section algorithm for pseudo-monotone variational inequalities.

Step 0: For given $p^0, p^1 \in C, \lambda_0 > 0, \overline{\lambda} > 0, \phi \in (1, \varphi],$$\varphi$ is, golden section ratio constant $\dfrac{\sqrt{5}+1}{2}.$

Let $\overline{p}^0 = p^1, \theta_0 = 1, \rho = \dfrac{1}{\phi}+\dfrac{1}{\phi^2}, k = 1$.

Step 1: Compute $x^k = A(p^k), x^{k-1} = A(p^{k-1}),$

Step 2: Compute

Step 3: Compute

Set $k: = k+1$, and go to Step 1.

Remark 4.1. Two key inequalities can be obtained from (4.1). First, we have $\lambda_k \leq (\frac{1}{\phi} + \frac{1}{\phi^2})\lambda_{k-1}$, it means $\theta_k -1-\frac{1}{\phi} \leq 0$. Second, combining $\frac{\phi\theta_{k-1}}{\lambda_{k-1}} = \frac{\theta_k \theta_{k-1}}{\lambda_k}$ and (4.1), we can gain

Then we have

Lemma 4.1. Suppose that the mapping $A:H \to H$ is $L-$ Lipschitz continuous. If the sequence $\{p^k\}$ generated by Algorithm 4.1 is bounded, then both $\{\lambda_k \}$ and $\{\theta_k\}$ are also bounded and separate from 0.

Proof. Obviously, $\{\lambda_k\}$ is decreasing and positive, so $\{\lambda_k\}$ is bounded. Next, the mathematical induction will be utilized to prove that $\{\lambda_k\}$ separates from 0.

Since $\{p^k\}$ is bounded, then there exists $L > 0$ such that

Set $L$ large enough, subject to $\lambda_i \geq \frac{\phi^2}{4L^2\overline{\lambda}}(i = 0, 1)$. Suppose that for all $i = 0, 1, \ldots, k-1$, $\lambda_i \geq \frac{\phi^2}{4L^2\overline{\lambda}}$ holds, then one has

or

So regardless of what the expression for $\{\lambda_k\}$is we have $\lambda_k \geq \frac{\phi^2}{4L^2\overline{\lambda}}$ holds, hence $\{\lambda_k\}$ separates from 0. While $\theta_k = \frac{\lambda_k}{\lambda_{k-1}} \phi$, by the similar method, itus easy to figure out that $\{\theta_k\}$ is also bounded and separates from 0.

To simplify the proof that follows, we define $\Psi(u, v) = \langle u^*, v-u \rangle$, and $u^* = A(u)$.

Theorem 4.1. Suppose that $A:H \to H$ is pseudo-monotone and satisfies $L-Lipschitz$ continuity. For every $p^1, \overline{p}^0 \in C, \lambda \in (0, \frac{\varphi}{2L}]$, the sequences $\{p^k\}$ and $\{\overline{p}^k\}$ generated by Algorithm 4.1 converge to $VI(C, A)$.

Proof. Combining $p^{k+1} = P_{C}(\overline{p}^k - \lambda_k x^k)$ and Lemma 2.1, we have

Set $k = k-1$ in (4.3), and take $z = p^{k+1}$, one obtains

Multiply both sides of (4.4) with $\frac{\lambda_k}{\lambda_{k-1}}$, and utilize the equation $\frac{\lambda_k}{\lambda_{k-1}}(p^k-\overline{p}^{k-1}) = \theta_k(p^k-\overline{p}^k)$; it infers

Combining (4.3) and (4.5), we obtain

Writing the first two terms of (4.6) in the form of a norm, we can obtain

According to $\overline{p}^{k+1} = \frac{(\phi-1)p^{k+1}+\overline{p}^k}{\phi}$, we have

The above equation combined with (4.7), one has

While $\theta_k \leq 1+\frac{1}{\phi}$, it follows from (4.2) that the rightmost term in (4.8) can be scaled to

However, after scaling (4.8) with this formula, it infers

After iteratively accumulating the above equation, we have

Set $z = p \in VI(C, A)$, then the last term on the left in (4.10) is nonnegative. It can gain $\{\overline{p}^k\}$ and $\{p^k\}$ are bounded, and $\theta_k||p^k-\overline{p}^k|| \to 0$. According to the proof of Lemma 4.1, we have $\lambda_k \geq \frac{\phi^2}{4L^2\overline{\lambda}}$ and $\{\theta_k\}$ separates from 0. Hence, $\lim\limits_{k \to +\infty}||p^k-\overline{p}^k|| = 0$ holds. It means that $p^k-\overline{p}^{k-1} \to 0$ and $p^{k+1}-p^k \to 0.$

Next, we will prove that every cluster point of $\{p^k\}$ and $\{\overline{p}^k\}$ belongs to $VI(C, A)$.

Set subsequence $\{k_i\}$ subjects to $p^{k_i} \to \widetilde{p}$ and $\lambda_{k_i} \to \lambda > 0(i \to +\infty)$ holds. Obviously, both $p^{k_i +1} \to \widetilde{p}$ and $\overline{p}^{k_i} \to \widetilde{p}$ hold. Substituting $k = k_i$ into (4.3), as $i$ increases to infinity, we have

Hence, $\widetilde{p} \in VI(C, A).$ According to (4.9), we can get $\frac{\phi}{\phi-1}||\overline{p}^k-z||^2+\frac{\theta_{k-1}}{2}||p^k-p^{k-1}||^2$ is non-monotonically increasing, which implies $\lim\limits_{k \to +\infty}||\overline{p}^k-z||$ exists. Since $z \in VI(C, A)$ is arbitrary, it follows from Lemma 2.9 states that there exist sequences $\{p^k\}$ and $\{\overline{p}^k\}$ converging to the certain elements of $VI(C, A)$. The proof is completed.

5.   Numerical experiments

In order to evaluate the performance of the proposed algorithms, we present numerical experiments relative to the variational inequality. In this section, we provide an example to compare Algorithms 3.1 and 4.1 with the modified projection and contraction algorithm ^[28], see Figure 1.

Example 5.1. Let $f :{\mathbb{R}}^n \rightarrow {\mathbb{R}}^n,$ be defined by $f(x) = Ax+b$, where $A = Z^T Z, Z = ({z_{ij}})_{n\times n}$ and $b = (b_i) \in {\mathbb{R}}^n$ where $z_{ij} \in [1, 100]$ and $b_i \in [-100, 0]$ are generated randomly.

Under the given parameters, we can obtain the convergence rate ratio of Algorithms 3.1 and 4.1 which is faster than Modified PCA.

Remark 5.1. First, we begin with a comparison of the stochastic projection and contraction algorithm in ^[29] and the fast alternated inertial projection algorithms in ^[30]. But the variational inequality conditions of these two literatures are too different from those of this literature, so I choose the literature ^[28] with more similar conditions and conclusions to compare.

6.   Conclusions

In this paper, the approximation problem has been investigated for the pseudo-monotonic variational inequalities. By taking advantage of the reciprocal form of parameters and the golden section ratio, two improved extrapolated gradient algorithms have been proposed, in which the strong convergence of the two defined algorithms for solving pseudo-monotone variational inequalities problems is achieved by extending the existing results of Thong ^[17] and Malitsky ^[21]. Finally, one numerical example is provided to illustrate the effectiveness of the proposed results.

Author contribution

Haoran Tang: Formal analysis, Writing editing, Visualization, Data curation; Weiqiang Gong: Funding acquisition, Visualization, Conceptualization, Supervision. All the authors have agreed and given their consent for the publication of this research paper.

Acknowledgments

The author is supported by the National Natural Science Foundation of China (Program No. 79970017) and the National Natural Science Foundation of China (Program No. 61906084).

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The author declares there is no conflict of interest.